Section 3.2 Relations 165 SECTION 3.2 Relations ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Express a relation in various forms and determine its domain and range. 2. Determine the domain and range from the graph of a relation. 3. Determine input and output values for a relation. 4. Troubleshoot common errors. Mathematicians, scientists, economists, and others are intrigued by the relations that exist between two quantities. They might study questions such as these. How does drinking alcohol affect a person’s response time? How does cholesterol intake affect the likelihood of a heart attack? How does study time affect a grade in a course? How do years of education affect income? In this section, we will study this concept of relations in more detail. Relations When we look up the definition of a relation, we find that almost every description involves the words connection and/or association. This idea extends to the definition of a mathematical relation. A mathematical relation describes the connection or association between two sets of information. This description is denoted by a set of ordered pairs. We use relations every day. When we checkout at the grocery store, each item scanned corresponds to a specific price. When a topic in Google is entered, search results are displayed. When we search a class schedule via the Web, we enter a specific course and are then given a list of all the classes for that course. In each of these cases, we pair together a piece of information from one set of values to a piece of information from another set of values. The first piece of information is the input value, or x-value. The second piece of information is the output value, or y-value. This set of ordered pairs is called a relation. Definition: A relation is a set of ordered pairs, in which the first coordinate of the ordered pairs comes from a set called the domain and the second coordinate of the ordered pairs comes from a set called the range. The domain is the set of x-values, or input values, and the range is the set of y-values, or output values. Relations can be expressed in various forms: a table, a mapping, a graph, an equation, and a set of ordered pairs. Table Mapping Graph Equation x y 0 15,000 1 12,500 2 10,000 3 7,500 4 5,000 5 2,500 6 0 Student Credit Hours 3 6 12 15 6 2 –4 –2 2 x 4 4 –2 y y = 0.24x + 4.42 Recall that an equation represents a relation because an equation defines a set of ordered pair solutions. When we state the domain or range of a relation, it is not necessary to list values more than once. Set of ordered pairs {(1, 3), (2, 4), (3, 5), (4, 6)} Objective 1 ▶ Express a relation in various forms and determine its domain and range.
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